All depends on how much access you have.
In some older homes you can gain access behind wooden baseboard to run wire.
If this room is above a basement you could drill up between the wall and fish the wires without cutting any sheetrock.
A right angle drill might work to drill thru the studs.
You could also cut out a rectangle section of drywall between the two studs for easiest access.
If you cut it neatly, it should go back without much fuss
There are other things to consider besides running wire.
Obviously this outlet feeds other devices, and as i understand, you want to use the power from this outlet to feed the switch.
You will need to determine which wire is the line in and which is the feeder.
When you find which wire is the source, twist it together with the new black wire and a pigtail.
You should also consider box fill limitations, wire size, and breaker amps.
If its 14awg on a 15amp breaker, youll probably be ok.